How do you find square root of 138?

1 Answer
Apr 3, 2018

Answer:

#sqrt(138) ~~ 39019777/3321584 ~~ 11.74734012447073#

Explanation:

The prime factorisation of #138# is:

#138 = 2 * 3 * 23#

Since this contains no squared terms, the square root cannot be simplified and not being a perfect square, it is irrational.

Note that:

#11^2 = 121 < 138 < 144 = 12^2#

So #sqrt(138)# is somewhere between #11# and #12#, closer to #12#.

Let us approximate it as #11 3/4 = 47/4#.

This is actually a very efficient approximation, since:

#47^2 = 2209 = 2208 + 1 = 4^2 * 138 + 1#

A much more formal way to find such an efficient initial approximation is to be found at https://socratic.org/s/aPLdnFSE

Next, consider the quadratic with zeros #47+4sqrt(138)# and #47-4sqrt(138)#...

#(x-47-4sqrt(138))(x-47+4sqrt(138)) = x^2-94x+1#

From this quadratic we can define an integer sequence recursively, using the rules:

#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 94a_(n+1)-a_n) :}#

The first few terms of this sequence are:

#0, 1, 94, 8835, 830396, 78048389#

The ratio between consecutive terms converges very rapidly towards #47+4sqrt(138)#

So we can approximate:

#sqrt(138) ~~ 1/4(78048389/830396-47)#

#color(white)(sqrt(138)) = 1/4(39019777/830396)#

#color(white)(sqrt(138)) = 39019777/3321584#

#color(white)(sqrt(138)) ~~ 11.74734012447073#